What is the purpose of the noise injection in the generator network of a GAN?
Your goal is to model a distribution when constructing a GAN, therefore you need a way to be able to sample that distribution. The noise's purpose is so you can do this. Generally, it's drawn from a distribution that is computationally easy to draw from (like a gaussian).   

You are modeling the generator $G(X)$  where $X \sim N(\mu, \sigma^2)$. this means $G(X)$ is a random variable itself. The forward pass of the network transforms the $X$ samples into our $G(X)$ samples, allowing us to formulate a loss function (by solving the expectation as the mean of drawn samples) and train the model.  

Takeaway: The noise injected is just a parametrization of our Generator in another space, and the training goal is to learn the ideal transformation (we use neural networks because they are differentiable and are effective function approximators)  

Also, note to your point of why it doesn't learn the training data (in its entirety) exactly is because generally $G(X)$ is continuous, and therefore if it has 2 images in its codomain, there also exists some path in pixel space from one to the other containing an uncountable (or quantized countable) number of images that don't exist in the training, and this would be reflected in the loss, therefore in the min-max game of the optimization, its difficult for it learn the training set on the nose.